This invention relates to a method for the exposive acceleration of at least one preformed fragment (or projectile) wherein the shock wave front triggered upon ignition of the explosive is introduced from the rear (with respect to the desired flight direction of the fragment) or from the rear and the side into the fragment, as well as to a warhead wherein preformed fragments are in contact with or embedded in an explosive charge in such a way that, upon detonation of the explosive charge, each fragment can be directly accelerated, and to the use of the warhead for the explosive acceleration of such fragments or projectiles and to the use as an underwater fragmentation warhead. The direct explosive acceleration of one or several preformed fragments has been known e.g. from DE 2,821,723 C2. In this reference a shock wave front passes from the rear (with respect to the flight direction) out of the explosive charge into the fragments. During this transition the shock wave front is hardly attenuated and it travels through the full length of the fragment until it is finally reflected from the forward surface of the fragment back into the projectile. This process leads to an undesired plastic strain of the fragment and usually results in the destruction of the projectile: the forward section is torn off. The ballistics of the remaining pieces of the projectile are unpredictable thus degrading the use as a warhead especially for underwater applications. Failure of the material under the influence of reflected shock waves occurs particularly in fragments with a ratio of length to maximum diameter of more than 3. Therefore, all of the fragments known heretofore that can be directly accelerated by an explosive charge predominantly exhibit a spherical shape or are formed irregularly without any preferred direction. On the other hand, the range of the ballistic flight of the fragment, especially in water, strongly depends on its length to maximum diameter ratio; the range increases with this ratio.